# HG changeset patch
# User oheimb
# Date 981057108 -3600
# Node ID a70b796d9af80d3cff43b1afb88f2a4a8ec1f874
# Parent  23bf8d787b04f4f3f810190b1be3c9e323860fb8
converted to Isar therory, adding attributes complete_split and split_format

diff -r 23bf8d787b04 -r a70b796d9af8 src/HOL/Induct/LList.ML
--- a/src/HOL/Induct/LList.ML	Thu Feb 01 20:51:13 2001 +0100
+++ b/src/HOL/Induct/LList.ML	Thu Feb 01 20:51:48 2001 +0100
@@ -782,7 +782,7 @@
 by (ALLGOALS Asm_simp_tac);
 qed "fun_power_Suc";
 
-val Pair_cong = read_instantiate_sg (sign_of Product_Type.thy)
+val Pair_cong = read_instantiate_sg (sign_of (theory "Product_Type"))
  [("f","Pair")] (standard(refl RS cong RS cong));
 
 (*The bisimulation consists of {(lmap(f)^n (h(u)), lmap(f)^n (iterates(f,u)))}
diff -r 23bf8d787b04 -r a70b796d9af8 src/HOL/Modelcheck/MuckeSyn.ML
--- a/src/HOL/Modelcheck/MuckeSyn.ML	Thu Feb 01 20:51:13 2001 +0100
+++ b/src/HOL/Modelcheck/MuckeSyn.ML	Thu Feb 01 20:51:48 2001 +0100
@@ -147,7 +147,7 @@
 
 (* first simplification, then model checking *)
 
-goalw Product_Type.thy [split_def] "(f::'a*'b=>'c) = (%(x, y). f (x, y))";
+goalw (theory "Product_Type") [split_def] "(f::'a*'b=>'c) = (%(x, y). f (x, y))";
   by (rtac ext 1);
   by (stac (surjective_pairing RS sym) 1);
   by (rtac refl 1);
diff -r 23bf8d787b04 -r a70b796d9af8 src/HOL/Product_Type.thy
--- a/src/HOL/Product_Type.thy	Thu Feb 01 20:51:13 2001 +0100
+++ b/src/HOL/Product_Type.thy	Thu Feb 01 20:51:48 2001 +0100
@@ -7,7 +7,11 @@
 The unit type.
 *)
 
-Product_Type = Fun +
+theory Product_Type = Fun
+files 
+  ("Tools/split_rule.ML")
+  ("Product_Type_lemmas.ML")
+:
 
 
 (** products **)
@@ -15,31 +19,34 @@
 (* type definition *)
 
 constdefs
-  Pair_Rep      :: ['a, 'b] => ['a, 'b] => bool
-  "Pair_Rep == (%a b. %x y. x=a & y=b)"
+  Pair_Rep :: "['a, 'b] => ['a, 'b] => bool"
+ "Pair_Rep == (%a b. %x y. x=a & y=b)"
 
 global
 
 typedef (Prod)
   ('a, 'b) "*"          (infixr 20)
     = "{f. ? a b. f = Pair_Rep (a::'a) (b::'b)}"
+proof
+  fix a b show "Pair_Rep a b : ?Prod"
+    by blast
+qed
 
 syntax (symbols)
-  "*"           :: [type, type] => type         ("(_ \\<times>/ _)" [21, 20] 20)
-
+  "*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)
 syntax (HTML output)
-  "*"           :: [type, type] => type         ("(_ \\<times>/ _)" [21, 20] 20)
+  "*"      :: "[type, type] => type"         ("(_ \<times>/ _)" [21, 20] 20)
 
 
 (* abstract constants and syntax *)
 
 consts
-  fst           :: "'a * 'b => 'a"
-  snd           :: "'a * 'b => 'b"
-  split         :: "[['a, 'b] => 'c, 'a * 'b] => 'c"
-  prod_fun      :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd"
-  Pair          :: "['a, 'b] => 'a * 'b"
-  Sigma         :: "['a set, 'a => 'b set] => ('a * 'b) set"
+  fst      :: "'a * 'b => 'a"
+  snd      :: "'a * 'b => 'b"
+  split    :: "[['a, 'b] => 'c, 'a * 'b] => 'c"
+  prod_fun :: "['a => 'b, 'c => 'd, 'a * 'c] => 'b * 'd"
+  Pair     :: "['a, 'b] => 'a * 'b"
+  Sigma    :: "['a set, 'a => 'b set] => ('a * 'b) set"
 
 
 (* patterns -- extends pre-defined type "pttrn" used in abstractions *)
@@ -51,11 +58,11 @@
   "_tuple"      :: "'a => tuple_args => 'a * 'b"        ("(1'(_,/ _'))")
   "_tuple_arg"  :: "'a => tuple_args"                   ("_")
   "_tuple_args" :: "'a => tuple_args => tuple_args"     ("_,/ _")
-  "_pattern"    :: [pttrn, patterns] => pttrn           ("'(_,/ _')")
-  ""            :: pttrn => patterns                    ("_")
-  "_patterns"   :: [pttrn, patterns] => patterns        ("_,/ _")
-  "@Sigma"      :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"   ("(3SIGMA _:_./ _)" 10)
-  "@Times"      :: "['a set, 'a => 'b set] => ('a * 'b) set"    (infixr "<*>" 80)
+  "_pattern"    :: "[pttrn, patterns] => pttrn"         ("'(_,/ _')")
+  ""            :: "pttrn => patterns"                  ("_")
+  "_patterns"   :: "[pttrn, patterns] => patterns"      ("_,/ _")
+  "@Sigma" ::"[pttrn, 'a set, 'b set] => ('a * 'b) set" ("(3SIGMA _:_./ _)" 10)
+  "@Times" ::"['a set,  'a => 'b set] => ('a * 'b) set" (infixr "<*>" 80)
 
 translations
   "(x, y)"       == "Pair x y"
@@ -70,8 +77,10 @@
   "A <*> B"      => "Sigma A (_K B)"
 
 syntax (symbols)
-  "@Sigma"      :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"   ("(3\\<Sigma> _\\<in>_./ _)" 10)
-  "@Times"      :: "['a set, 'a => 'b set] => ('a * 'b) set"    ("_ \\<times> _" [81, 80] 80)
+  "@Sigma" :: "[pttrn, 'a set, 'b set] => ('a * 'b) set"  ("(3\<Sigma> _\<in>_./ _)"   10)
+  "@Times" :: "['a set,  'a => 'b set] => ('a * 'b) set"  ("_ \<times> _" [81, 80] 80)
+
+print_translation {* [("Sigma", dependent_tr' ("@Sigma", "@Times"))]; *}
 
 
 (* definitions *)
@@ -79,12 +88,12 @@
 local
 
 defs
-  Pair_def      "Pair a b == Abs_Prod(Pair_Rep a b)"
-  fst_def       "fst p == @a. ? b. p = (a, b)"
-  snd_def       "snd p == @b. ? a. p = (a, b)"
-  split_def     "split == (%c p. c (fst p) (snd p))"
-  prod_fun_def  "prod_fun f g == split(%x y.(f(x), g(y)))"
-  Sigma_def     "Sigma A B == UN x:A. UN y:B(x). {(x, y)}"
+  Pair_def:     "Pair a b == Abs_Prod(Pair_Rep a b)"
+  fst_def:      "fst p == @a. ? b. p = (a, b)"
+  snd_def:      "snd p == @b. ? a. p = (a, b)"
+  split_def:    "split == (%c p. c (fst p) (snd p))"
+  prod_fun_def: "prod_fun f g == split(%x y.(f(x), g(y)))"
+  Sigma_def:    "Sigma A B == UN x:A. UN y:B(x). {(x, y)}"
 
 
 
@@ -92,7 +101,11 @@
 
 global
 
-typedef  unit = "{True}"
+typedef  unit = "{True}" 
+proof
+  show "True : ?unit"
+    by blast
+qed
 
 consts
   "()"          :: unit                           ("'(')")
@@ -100,10 +113,11 @@
 local
 
 defs
-  Unity_def     "() == Abs_unit True"
+  Unity_def:    "() == Abs_unit True"
+
+use "Product_Type_lemmas.ML"
+
+use "Tools/split_rule.ML"
+setup split_attributes
 
 end
-
-ML
-
-val print_translation = [("Sigma", dependent_tr' ("@Sigma", "@Times"))];
diff -r 23bf8d787b04 -r a70b796d9af8 src/HOL/Product_Type_lemmas.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Product_Type_lemmas.ML	Thu Feb 01 20:51:48 2001 +0100
@@ -0,0 +1,584 @@
+(*  Title:      HOL/Product_Type_lemmas.ML
+    ID:         $Id$
+    Author:     Lawrence C Paulson, Cambridge University Computer Laboratory
+    Copyright   1991  University of Cambridge
+
+Ordered Pairs, the Cartesian product type, the unit type
+*)
+
+(* ML bindings *)
+val Pair_def  = thm "Pair_def";
+val fst_def   = thm "fst_def";
+val snd_def   = thm "snd_def";
+val split_def = thm "split_def";
+val prod_fun_def = thm "prod_fun_def";
+val Sigma_def = thm "Sigma_def";
+val Unity_def = thm "Unity_def";
+
+
+(** unit **)
+
+Goalw [Unity_def] "u = ()";
+by (stac (rewrite_rule [thm"unit_def"] (thm"Rep_unit") RS singletonD RS sym) 1);
+by (rtac (thm "Rep_unit_inverse" RS sym) 1);
+qed "unit_eq";
+
+(*simplification procedure for unit_eq.
+  Cannot use this rule directly -- it loops!*)
+local
+  val unit_pat = Thm.cterm_of (Theory.sign_of (the_context ())) (Free ("x", HOLogic.unitT));
+  val unit_meta_eq = standard (mk_meta_eq unit_eq);
+  fun proc _ _ t =
+    if HOLogic.is_unit t then None
+    else Some unit_meta_eq;
+in
+  val unit_eq_proc = Simplifier.mk_simproc "unit_eq" [unit_pat] proc;
+end;
+
+Addsimprocs [unit_eq_proc];
+
+Goal "(!!x::unit. PROP P x) == PROP P ()";
+by (Simp_tac 1);
+qed "unit_all_eq1";
+
+Goal "(!!x::unit. PROP P) == PROP P";
+by (rtac triv_forall_equality 1);
+qed "unit_all_eq2";
+
+Goal "P () ==> P x";
+by (Simp_tac 1);
+qed "unit_induct";
+
+(*This rewrite counters the effect of unit_eq_proc on (%u::unit. f u),
+  replacing it by f rather than by %u.f(). *)
+Goal "(%u::unit. f()) = f";
+by (rtac ext 1);
+by (Simp_tac 1);
+qed "unit_abs_eta_conv";
+Addsimps [unit_abs_eta_conv];
+
+
+(** prod **)
+
+Goalw [thm "Prod_def"] "Pair_Rep a b : Prod";
+by (EVERY1 [rtac CollectI, rtac exI, rtac exI, rtac refl]);
+qed "ProdI";
+
+Goalw [thm "Pair_Rep_def"] "Pair_Rep a b = Pair_Rep a' b' ==> a=a' & b=b'";
+by (dtac (fun_cong RS fun_cong) 1);
+by (Blast_tac 1);
+qed "Pair_Rep_inject";
+
+Goal "inj_on Abs_Prod Prod";
+by (rtac inj_on_inverseI 1);
+by (etac (thm "Abs_Prod_inverse") 1);
+qed "inj_on_Abs_Prod";
+
+val prems = Goalw [Pair_def]
+    "[| (a, b) = (a',b');  [| a=a';  b=b' |] ==> R |] ==> R";
+by (rtac (inj_on_Abs_Prod RS inj_onD RS Pair_Rep_inject RS conjE) 1);
+by (REPEAT (ares_tac (prems@[ProdI]) 1));
+qed "Pair_inject";
+
+Goal "((a,b) = (a',b')) = (a=a' & b=b')";
+by (blast_tac (claset() addSEs [Pair_inject]) 1);
+qed "Pair_eq";
+AddIffs [Pair_eq];
+
+Goalw [fst_def] "fst (a,b) = a";
+by (Blast_tac 1);
+qed "fst_conv";
+Goalw [snd_def] "snd (a,b) = b";
+by (Blast_tac 1);
+qed "snd_conv";
+Addsimps [fst_conv, snd_conv];
+
+Goal "fst (x, y) = a ==> x = a";
+by (Asm_full_simp_tac 1);
+qed "fst_eqD";
+Goal "snd (x, y) = a ==> y = a";
+by (Asm_full_simp_tac 1);
+qed "snd_eqD";
+
+Goalw [Pair_def] "? x y. p = (x,y)";
+by (rtac (rewrite_rule [thm "Prod_def"] (thm "Rep_Prod") RS CollectE) 1);
+by (EVERY1[etac exE, etac exE, rtac exI, rtac exI,
+           rtac (thm "Rep_Prod_inverse" RS sym RS trans),  etac arg_cong]);
+qed "PairE_lemma";
+
+val [prem] = Goal "[| !!x y. p = (x,y) ==> Q |] ==> Q";
+by (rtac (PairE_lemma RS exE) 1);
+by (REPEAT (eresolve_tac [prem,exE] 1));
+qed "PairE";
+
+fun pair_tac s = EVERY' [res_inst_tac [("p",s)] PairE, hyp_subst_tac,
+                         K prune_params_tac];
+
+(* Do not add as rewrite rule: invalidates some proofs in IMP *)
+Goal "p = (fst(p),snd(p))";
+by (pair_tac "p" 1);
+by (Asm_simp_tac 1);
+qed "surjective_pairing";
+Addsimps [surjective_pairing RS sym];
+
+Goal "? x y. z = (x, y)";
+by (rtac exI 1);
+by (rtac exI 1);
+by (rtac surjective_pairing 1);
+qed "surj_pair";
+Addsimps [surj_pair];
+
+
+bind_thm ("split_paired_all",
+  SplitPairedAll.rule (standard (surjective_pairing RS eq_reflection)));
+bind_thms ("split_tupled_all", [split_paired_all, unit_all_eq2]);
+
+(*
+Addsimps [split_paired_all] does not work with simplifier
+because it also affects premises in congrence rules,
+where is can lead to premises of the form !!a b. ... = ?P(a,b)
+which cannot be solved by reflexivity.
+*)
+
+(* replace parameters of product type by individual component parameters *)
+local
+  fun exists_paired_all (Const ("all", _) $ Abs (_, T, t)) =
+        can HOLogic.dest_prodT T orelse exists_paired_all t
+    | exists_paired_all (t $ u) = exists_paired_all t orelse exists_paired_all u
+    | exists_paired_all (Abs (_, _, t)) = exists_paired_all t
+    | exists_paired_all _ = false;
+  val ss = HOL_basic_ss
+    addsimps [split_paired_all, unit_all_eq2, unit_abs_eta_conv]
+    addsimprocs [unit_eq_proc];
+in
+  val split_all_tac = SUBGOAL (fn (t, i) =>
+    if exists_paired_all t then full_simp_tac ss i else no_tac);
+  fun split_all th =
+    if exists_paired_all (#prop (Thm.rep_thm th)) then full_simplify ss th else th;
+end;
+
+claset_ref() := claset()
+  addSWrapper ("split_all_tac", fn tac2 => split_all_tac ORELSE' tac2);
+
+Goal "(!x. P x) = (!a b. P(a,b))";
+by (Fast_tac 1);
+qed "split_paired_All";
+Addsimps [split_paired_All];
+(* AddIffs is not a good idea because it makes Blast_tac loop *)
+
+bind_thm ("prod_induct",
+  allI RS (allI RS (split_paired_All RS iffD2)) RS spec);
+
+Goal "(? x. P x) = (? a b. P(a,b))";
+by (Fast_tac 1);
+qed "split_paired_Ex";
+Addsimps [split_paired_Ex];
+
+Goalw [split_def] "split c (a,b) = c a b";
+by (Simp_tac 1);
+qed "split_conv";
+Addsimps [split_conv];
+(*bind_thm ("split", split_conv);                  (*for compatibility*)*)
+
+(*Subsumes the old split_Pair when f is the identity function*)
+Goal "split (%x y. f(x,y)) = f";
+by (rtac ext 1);
+by (pair_tac "x" 1);
+by (Simp_tac 1);
+qed "split_Pair_apply";
+
+(*Can't be added to simpset: loops!*)
+Goal "(SOME x. P x) = (SOME (a,b). P(a,b))";
+by (simp_tac (simpset() addsimps [split_Pair_apply]) 1);
+qed "split_paired_Eps";
+
+Goalw [split_def] "Eps (split P) = (SOME xy. P (fst xy) (snd xy))";
+by (rtac refl 1);
+qed "Eps_split";
+
+Goal "!!s t. (s=t) = (fst(s)=fst(t) & snd(s)=snd(t))";
+by (split_all_tac 1);
+by (Asm_simp_tac 1);
+qed "Pair_fst_snd_eq";
+
+Goal "fst p = fst q ==> snd p = snd q ==> p = q";
+by (asm_simp_tac (simpset() addsimps [Pair_fst_snd_eq]) 1);
+qed "prod_eqI";
+AddXIs [prod_eqI];
+
+(*Prevents simplification of c: much faster*)
+Goal "p=q ==> split c p = split c q";
+by (etac arg_cong 1);
+qed "split_weak_cong";
+
+Goal "(%(x,y). f(x,y)) = f";
+by (rtac ext 1);
+by (split_all_tac 1);
+by (rtac split_conv 1);
+qed "split_eta";
+
+val prems = Goal "(!!x y. f x y = g(x,y)) ==> (%(x,y). f x y) = g";
+by (asm_simp_tac (simpset() addsimps prems@[split_eta]) 1);
+qed "cond_split_eta";
+
+(*simplification procedure for cond_split_eta.
+  using split_eta a rewrite rule is not general enough, and using
+  cond_split_eta directly would render some existing proofs very inefficient.
+  similarly for split_beta. *)
+local
+  fun  Pair_pat k 0 (Bound m) = (m = k)
+  |    Pair_pat k i (Const ("Pair",  _) $ Bound m $ t) = i > 0 andalso
+                        m = k+i andalso Pair_pat k (i-1) t
+  |    Pair_pat _ _ _ = false;
+  fun no_args k i (Abs (_, _, t)) = no_args (k+1) i t
+  |   no_args k i (t $ u) = no_args k i t andalso no_args k i u
+  |   no_args k i (Bound m) = m < k orelse m > k+i
+  |   no_args _ _ _ = true;
+  fun split_pat tp i (Abs  (_,_,t)) = if tp 0 i t then Some (i,t) else None
+  |   split_pat tp i (Const ("split", _) $ Abs (_, _, t)) = split_pat tp (i+1) t
+  |   split_pat tp i _ = None;
+  fun metaeq sg lhs rhs = mk_meta_eq (prove_goalw_cterm []
+        (cterm_of sg (HOLogic.mk_Trueprop (HOLogic.mk_eq (lhs,rhs))))
+        (K [simp_tac (HOL_basic_ss addsimps [cond_split_eta]) 1]));
+  val sign = sign_of (the_context ());
+  fun simproc name patstr = Simplifier.mk_simproc name
+                [Thm.read_cterm sign (patstr, HOLogic.termT)];
+
+  val beta_patstr = "split f z";
+  val  eta_patstr = "split f";
+  fun beta_term_pat k i (Abs (_, _, t)) = beta_term_pat (k+1) i t
+  |   beta_term_pat k i (t $ u) = Pair_pat k i (t $ u) orelse
+                        (beta_term_pat k i t andalso beta_term_pat k i u)
+  |   beta_term_pat k i t = no_args k i t;
+  fun  eta_term_pat k i (f $ arg) = no_args k i f andalso Pair_pat k i arg
+  |    eta_term_pat _ _ _ = false;
+  fun subst arg k i (Abs (x, T, t)) = Abs (x, T, subst arg (k+1) i t)
+  |   subst arg k i (t $ u) = if Pair_pat k i (t $ u) then incr_boundvars k arg
+                              else (subst arg k i t $ subst arg k i u)
+  |   subst arg k i t = t;
+  fun beta_proc sg _ (s as Const ("split", _) $ Abs (_, _, t) $ arg) =
+        (case split_pat beta_term_pat 1 t of
+        Some (i,f) => Some (metaeq sg s (subst arg 0 i f))
+        | None => None)
+  |   beta_proc _ _ _ = None;
+  fun eta_proc sg _ (s as Const ("split", _) $ Abs (_, _, t)) =
+        (case split_pat eta_term_pat 1 t of
+          Some (_,ft) => Some (metaeq sg s (let val (f $ arg) = ft in f end))
+        | None => None)
+  |   eta_proc _ _ _ = None;
+in
+  val split_beta_proc = simproc "split_beta" beta_patstr beta_proc;
+  val split_eta_proc  = simproc "split_eta"   eta_patstr  eta_proc;
+end;
+
+Addsimprocs [split_beta_proc,split_eta_proc];
+
+Goal "(%(x,y). P x y) z = P (fst z) (snd z)";
+by (stac surjective_pairing 1 THEN rtac split_conv 1);
+qed "split_beta";
+
+(*For use with split_tac and the simplifier*)
+Goal "R (split c p) = (! x y. p = (x,y) --> R (c x y))";
+by (stac surjective_pairing 1);
+by (stac split_conv 1);
+by (Blast_tac 1);
+qed "split_split";
+
+(* could be done after split_tac has been speeded up significantly:
+simpset_ref() := simpset() addsplits [split_split];
+   precompute the constants involved and don't do anything unless
+   the current goal contains one of those constants
+*)
+
+Goal "R (split c p) = (~(? x y. p = (x,y) & (~R (c x y))))";
+by (stac split_split 1);
+by (Simp_tac 1);
+qed "split_split_asm";
+
+(** split used as a logical connective or set former **)
+
+(*These rules are for use with blast_tac.
+  Could instead call simp_tac/asm_full_simp_tac using split as rewrite.*)
+
+Goal "!!p. [| !!a b. p=(a,b) ==> c a b |] ==> split c p";
+by (split_all_tac 1);
+by (Asm_simp_tac 1);
+qed "splitI2";
+
+Goal "!!p. [| !!a b. (a,b)=p ==> c a b x |] ==> split c p x";
+by (split_all_tac 1);
+by (Asm_simp_tac 1);
+qed "splitI2'";
+
+Goal "c a b ==> split c (a,b)";
+by (Asm_simp_tac 1);
+qed "splitI";
+
+val prems = Goalw [split_def]
+    "[| split c p;  !!x y. [| p = (x,y);  c x y |] ==> Q |] ==> Q";
+by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1));
+qed "splitE";
+
+val prems = Goalw [split_def]
+    "[| split c p z;  !!x y. [| p = (x,y);  c x y z |] ==> Q |] ==> Q";
+by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1));
+qed "splitE'";
+
+val major::prems = Goal
+    "[| Q (split P z);  !!x y. [|z = (x, y); Q (P x y)|] ==> R  \
+\    |] ==> R";
+by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1));
+by (rtac (split_beta RS subst) 1 THEN rtac major 1);
+qed "splitE2";
+
+Goal "split R (a,b) ==> R a b";
+by (etac (split_conv RS iffD1) 1);
+qed "splitD";
+
+Goal "z: c a b ==> z: split c (a,b)";
+by (Asm_simp_tac 1);
+qed "mem_splitI";
+
+Goal "!!p. [| !!a b. p=(a,b) ==> z: c a b |] ==> z: split c p";
+by (split_all_tac 1);
+by (Asm_simp_tac 1);
+qed "mem_splitI2";
+
+val prems = Goalw [split_def]
+    "[| z: split c p;  !!x y. [| p = (x,y);  z: c x y |] ==> Q |] ==> Q";
+by (REPEAT (resolve_tac (prems@[surjective_pairing]) 1));
+qed "mem_splitE";
+
+AddSIs [splitI, splitI2, splitI2', mem_splitI, mem_splitI2];
+AddSEs [splitE, splitE', mem_splitE];
+
+Goal "(%u. ? x y. u = (x, y) & P (x, y)) = P";
+by (rtac ext 1);
+by (Fast_tac 1);
+qed "split_eta_SetCompr";
+Addsimps [split_eta_SetCompr];
+
+Goal "(%u. ? x y. u = (x, y) & P x y) = split P";
+br ext 1;
+by (Fast_tac 1);
+qed "split_eta_SetCompr2";
+Addsimps [split_eta_SetCompr2];
+
+(* allows simplifications of nested splits in case of independent predicates *)
+Goal "(%(a,b). P & Q a b) = (%ab. P & split Q ab)";
+by (rtac ext 1);
+by (Blast_tac 1);
+qed "split_part";
+Addsimps [split_part];
+
+Goal "(@(x',y'). x = x' & y = y') = (x,y)";
+by (Blast_tac 1);
+qed "Eps_split_eq";
+Addsimps [Eps_split_eq];
+(*
+the following  would be slightly more general,
+but cannot be used as rewrite rule:
+### Cannot add premise as rewrite rule because it contains (type) unknowns:
+### ?y = .x
+Goal "[| P y; !!x. P x ==> x = y |] ==> (@(x',y). x = x' & P y) = (x,y)";
+by (rtac some_equality 1);
+by ( Simp_tac 1);
+by (split_all_tac 1);
+by (Asm_full_simp_tac 1);
+qed "Eps_split_eq";
+*)
+
+(*** prod_fun -- action of the product functor upon functions ***)
+
+Goalw [prod_fun_def] "prod_fun f g (a,b) = (f(a),g(b))";
+by (rtac split_conv 1);
+qed "prod_fun";
+Addsimps [prod_fun];
+
+Goal "prod_fun (f1 o f2) (g1 o g2) = ((prod_fun f1 g1) o (prod_fun f2 g2))";
+by (rtac ext 1);
+by (pair_tac "x" 1);
+by (Asm_simp_tac 1);
+qed "prod_fun_compose";
+
+Goal "prod_fun (%x. x) (%y. y) = (%z. z)";
+by (rtac ext 1);
+by (pair_tac "z" 1);
+by (Asm_simp_tac 1);
+qed "prod_fun_ident";
+Addsimps [prod_fun_ident];
+
+Goal "(a,b):r ==> (f(a),g(b)) : (prod_fun f g)`r";
+by (rtac image_eqI 1);
+by (rtac (prod_fun RS sym) 1);
+by (assume_tac 1);
+qed "prod_fun_imageI";
+
+val major::prems = Goal
+    "[| c: (prod_fun f g)`r;  !!x y. [| c=(f(x),g(y));  (x,y):r |] ==> P  \
+\    |] ==> P";
+by (rtac (major RS imageE) 1);
+by (res_inst_tac [("p","x")] PairE 1);
+by (resolve_tac prems 1);
+by (Blast_tac 2);
+by (blast_tac (claset() addIs [prod_fun]) 1);
+qed "prod_fun_imageE";
+
+AddIs  [prod_fun_imageI];
+AddSEs [prod_fun_imageE];
+
+
+(*** Disjoint union of a family of sets - Sigma ***)
+
+Goalw [Sigma_def] "[| a:A;  b:B(a) |] ==> (a,b) : Sigma A B";
+by (REPEAT (ares_tac [singletonI,UN_I] 1));
+qed "SigmaI";
+
+AddSIs [SigmaI];
+
+(*The general elimination rule*)
+val major::prems = Goalw [Sigma_def]
+    "[| c: Sigma A B;  \
+\       !!x y.[| x:A;  y:B(x);  c=(x,y) |] ==> P \
+\    |] ==> P";
+by (cut_facts_tac [major] 1);
+by (REPEAT (eresolve_tac [UN_E, singletonE] 1 ORELSE ares_tac prems 1)) ;
+qed "SigmaE";
+
+(** Elimination of (a,b):A*B -- introduces no eigenvariables **)
+
+Goal "(a,b) : Sigma A B ==> a : A";
+by (etac SigmaE 1);
+by (REPEAT (eresolve_tac [asm_rl,Pair_inject,ssubst] 1)) ;
+qed "SigmaD1";
+
+Goal "(a,b) : Sigma A B ==> b : B(a)";
+by (etac SigmaE 1);
+by (REPEAT (eresolve_tac [asm_rl,Pair_inject,ssubst] 1)) ;
+qed "SigmaD2";
+
+val [major,minor]= Goal
+    "[| (a,b) : Sigma A B;    \
+\       [| a:A;  b:B(a) |] ==> P   \
+\    |] ==> P";
+by (rtac minor 1);
+by (rtac (major RS SigmaD1) 1);
+by (rtac (major RS SigmaD2) 1) ;
+qed "SigmaE2";
+
+AddSEs [SigmaE2, SigmaE];
+
+val prems = Goal
+    "[| A<=C;  !!x. x:A ==> B x <= D x |] ==> Sigma A B <= Sigma C D";
+by (cut_facts_tac prems 1);
+by (blast_tac (claset() addIs (prems RL [subsetD])) 1);
+qed "Sigma_mono";
+
+Goal "Sigma {} B = {}";
+by (Blast_tac 1) ;
+qed "Sigma_empty1";
+
+Goal "A <*> {} = {}";
+by (Blast_tac 1) ;
+qed "Sigma_empty2";
+
+Addsimps [Sigma_empty1,Sigma_empty2];
+
+Goal "UNIV <*> UNIV = UNIV";
+by Auto_tac;
+qed "UNIV_Times_UNIV";
+Addsimps [UNIV_Times_UNIV];
+
+Goal "- (UNIV <*> A) = UNIV <*> (-A)";
+by Auto_tac;
+qed "Compl_Times_UNIV1";
+
+Goal "- (A <*> UNIV) = (-A) <*> UNIV";
+by Auto_tac;
+qed "Compl_Times_UNIV2";
+
+Addsimps [Compl_Times_UNIV1, Compl_Times_UNIV2];
+
+Goal "((a,b): Sigma A B) = (a:A & b:B(a))";
+by (Blast_tac 1);
+qed "mem_Sigma_iff";
+AddIffs [mem_Sigma_iff];
+
+Goal "x:C ==> (A <*> C <= B <*> C) = (A <= B)";
+by (Blast_tac 1);
+qed "Times_subset_cancel2";
+
+Goal "x:C ==> (A <*> C = B <*> C) = (A = B)";
+by (blast_tac (claset() addEs [equalityE]) 1);
+qed "Times_eq_cancel2";
+
+Goal "Collect (split (%x y. P x & Q x y)) = (SIGMA x:Collect P. Collect (Q x))";
+by (Fast_tac 1);
+qed "SetCompr_Sigma_eq";
+
+(*** Complex rules for Sigma ***)
+
+Goal "{(a,b). P a & Q b} = Collect P <*> Collect Q";
+by (Blast_tac 1);
+qed "Collect_split";
+
+Addsimps [Collect_split];
+
+(*Suggested by Pierre Chartier*)
+Goal "(UN (a,b):(A <*> B). E a <*> F b) = (UNION A E) <*> (UNION B F)";
+by (Blast_tac 1);
+qed "UN_Times_distrib";
+
+Goal "(ALL z: Sigma A B. P z) = (ALL x:A. ALL y: B x. P(x,y))";
+by (Fast_tac 1);
+qed "split_paired_Ball_Sigma";
+Addsimps [split_paired_Ball_Sigma];
+
+Goal "(EX z: Sigma A B. P z) = (EX x:A. EX y: B x. P(x,y))";
+by (Fast_tac 1);
+qed "split_paired_Bex_Sigma";
+Addsimps [split_paired_Bex_Sigma];
+
+Goal "(SIGMA i:I Un J. C(i)) = (SIGMA i:I. C(i)) Un (SIGMA j:J. C(j))";
+by (Blast_tac 1);
+qed "Sigma_Un_distrib1";
+
+Goal "(SIGMA i:I. A(i) Un B(i)) = (SIGMA i:I. A(i)) Un (SIGMA i:I. B(i))";
+by (Blast_tac 1);
+qed "Sigma_Un_distrib2";
+
+Goal "(SIGMA i:I Int J. C(i)) = (SIGMA i:I. C(i)) Int (SIGMA j:J. C(j))";
+by (Blast_tac 1);
+qed "Sigma_Int_distrib1";
+
+Goal "(SIGMA i:I. A(i) Int B(i)) = (SIGMA i:I. A(i)) Int (SIGMA i:I. B(i))";
+by (Blast_tac 1);
+qed "Sigma_Int_distrib2";
+
+Goal "(SIGMA i:I - J. C(i)) = (SIGMA i:I. C(i)) - (SIGMA j:J. C(j))";
+by (Blast_tac 1);
+qed "Sigma_Diff_distrib1";
+
+Goal "(SIGMA i:I. A(i) - B(i)) = (SIGMA i:I. A(i)) - (SIGMA i:I. B(i))";
+by (Blast_tac 1);
+qed "Sigma_Diff_distrib2";
+
+Goal "Sigma (Union X) B = (UN A:X. Sigma A B)";
+by (Blast_tac 1);
+qed "Sigma_Union";
+
+(*Non-dependent versions are needed to avoid the need for higher-order
+  matching, especially when the rules are re-oriented*)
+Goal "(A Un B) <*> C = (A <*> C) Un (B <*> C)";
+by (Blast_tac 1);
+qed "Times_Un_distrib1";
+
+Goal "(A Int B) <*> C = (A <*> C) Int (B <*> C)";
+by (Blast_tac 1);
+qed "Times_Int_distrib1";
+
+Goal "(A - B) <*> C = (A <*> C) - (B <*> C)";
+by (Blast_tac 1);
+qed "Times_Diff_distrib1";
+
+
diff -r 23bf8d787b04 -r a70b796d9af8 src/HOL/Tools/split_rule.ML
--- /dev/null	Thu Jan 01 00:00:00 1970 +0000
+++ b/src/HOL/Tools/split_rule.ML	Thu Feb 01 20:51:48 2001 +0100
@@ -0,0 +1,95 @@
+(*Attempts to remove occurrences of split, and pair-valued parameters*)
+val remove_split = rewrite_rule [split_conv RS eq_reflection] o split_all;
+
+local
+
+(*In ap_split S T u, term u expects separate arguments for the factors of S,
+  with result type T.  The call creates a new term expecting one argument
+  of type S.*)
+fun ap_split (Type ("*", [T1, T2])) T3 u =
+      HOLogic.split_const (T1, T2, T3) $
+      Abs("v", T1,
+          ap_split T2 T3
+             ((ap_split T1 (HOLogic.prodT_factors T2 ---> T3) (incr_boundvars 1 u)) $
+              Bound 0))
+  | ap_split T T3 u = u;
+
+(*Curries any Var of function type in the rule*)
+fun split_rule_var' (t as Var (v, Type ("fun", [T1, T2])), rl) =
+      let val T' = HOLogic.prodT_factors T1 ---> T2
+          val newt = ap_split T1 T2 (Var (v, T'))
+          val cterm = Thm.cterm_of (#sign (rep_thm rl))
+      in
+          instantiate ([], [(cterm t, cterm newt)]) rl
+      end
+  | split_rule_var' (t, rl) = rl;
+
+(*** Complete splitting of partially splitted rules ***)
+
+fun ap_split' (T::Ts) U u = Abs ("v", T, ap_split' Ts U
+      (ap_split T (flat (map HOLogic.prodT_factors Ts) ---> U)
+        (incr_boundvars 1 u) $ Bound 0))
+  | ap_split' _ _ u = u;
+
+fun complete_split_rule_var ((t as Var (v, T), ts), (rl, vs)) =
+      let
+        val cterm = Thm.cterm_of (#sign (rep_thm rl))
+        val (Us', U') = strip_type T;
+        val Us = take (length ts, Us');
+        val U = drop (length ts, Us') ---> U';
+        val T' = flat (map HOLogic.prodT_factors Us) ---> U;
+        fun mk_tuple ((xs, insts), v as Var ((a, _), T)) =
+              let
+                val Ts = HOLogic.prodT_factors T;
+                val ys = variantlist (replicate (length Ts) a, xs);
+              in (xs @ ys, (cterm v, cterm (HOLogic.mk_tuple T
+                (map (Var o apfst (rpair 0)) (ys ~~ Ts))))::insts)
+              end
+          | mk_tuple (x, _) = x;
+        val newt = ap_split' Us U (Var (v, T'));
+        val cterm = Thm.cterm_of (#sign (rep_thm rl));
+        val (vs', insts) = foldl mk_tuple ((vs, []), ts);
+      in
+        (instantiate ([], [(cterm t, cterm newt)] @ insts) rl, vs')
+      end
+  | complete_split_rule_var (_, x) = x;
+
+fun collect_vars (vs, Abs (_, _, t)) = collect_vars (vs, t)
+  | collect_vars (vs, t) = (case strip_comb t of
+        (v as Var _, ts) => (v, ts)::vs
+      | (t, ts) => foldl collect_vars (vs, ts));
+
+in
+
+val split_rule_var = standard o remove_split o split_rule_var';
+
+(*Curries ALL function variables occurring in a rule's conclusion*)
+fun split_rule rl = standard (remove_split (foldr split_rule_var' (term_vars (concl_of rl), rl)));
+
+fun complete_split_rule rl =
+  standard (remove_split (fst (foldr complete_split_rule_var
+    (collect_vars ([], concl_of rl),
+     (rl, map (fst o fst o dest_Var) (term_vars (#prop (rep_thm rl))))))))
+	|> RuleCases.save rl;
+
+end;
+fun complete_split x = 
+	Attrib.no_args (Drule.rule_attribute (K complete_split_rule)) x;
+
+fun split_rule_goal xss rl = let 
+	val ss = HOL_basic_ss addsimps [split_conv, fst_conv, snd_conv]; 
+	fun one_split i (th,s) = rule_by_tactic (pair_tac s i) th;
+	fun one_goal (xs,i) th = foldl (one_split i) (th,xs);
+        in standard (Simplifier.full_simplify ss (foldln one_goal xss rl))
+	   |> RuleCases.save rl
+        end;
+fun split_format x = 
+	Attrib.syntax (Args.and_list1 (Scan.lift (Scan.repeat Args.name)) 
+	>> (fn xss => Drule.rule_attribute (K (split_rule_goal xss)))) x;
+
+val split_attributes = [Attrib.add_attributes 
+	[("complete_split", (complete_split, complete_split), 
+          "recursively split all pair-typed function arguments"),
+	 ("split_format", (split_format, split_format), 
+          "split given pair-typed subterms in premises")]];
+
diff -r 23bf8d787b04 -r a70b796d9af8 src/HOL/ex/cla.ML
--- a/src/HOL/ex/cla.ML	Thu Feb 01 20:51:13 2001 +0100
+++ b/src/HOL/ex/cla.ML	Thu Feb 01 20:51:48 2001 +0100
@@ -462,7 +462,8 @@
 
 (*From Davis, Obvious Logical Inferences, IJCAI-81, 530-531
   Fast_tac indeed copes!*)
-goal Product_Type.thy "(ALL x. F(x) & ~G(x) --> (EX y. H(x,y) & J(y))) & \
+goal (theory "Product_Type") 
+             "(ALL x. F(x) & ~G(x) --> (EX y. H(x,y) & J(y))) & \
 \             (EX x. K(x) & F(x) & (ALL y. H(x,y) --> K(y))) &   \
 \             (ALL x. K(x) --> ~G(x))  -->  (EX x. K(x) & J(x))";
 by (Fast_tac 1);
@@ -470,7 +471,7 @@
 
 (*From Rudnicki, Obvious Inferences, JAR 3 (1987), 383-393.  
   It does seem obvious!*)
-goal Product_Type.thy
+goal (theory "Product_Type")
     "(ALL x. F(x) & ~G(x) --> (EX y. H(x,y) & J(y))) &        \
 \    (EX x. K(x) & F(x) & (ALL y. H(x,y) --> K(y)))  &        \
 \    (ALL x. K(x) --> ~G(x))   -->   (EX x. K(x) --> ~G(x))";
@@ -488,7 +489,7 @@
 by (Blast_tac 1);
 result();
 
-goal Product_Type.thy
+goal (theory "Product_Type")
     "(ALL x y. R(x,y) | R(y,x)) &                \
 \    (ALL x y. S(x,y) & S(y,x) --> x=y) &        \
 \    (ALL x y. R(x,y) --> S(x,y))    -->   (ALL x y. S(x,y) --> R(x,y))";
diff -r 23bf8d787b04 -r a70b796d9af8 src/HOLCF/Cprod1.ML
--- a/src/HOLCF/Cprod1.ML	Thu Feb 01 20:51:13 2001 +0100
+++ b/src/HOLCF/Cprod1.ML	Thu Feb 01 20:51:48 2001 +0100
@@ -11,11 +11,12 @@
 (* less_cprod is a partial order on 'a * 'b                                 *)
 (* ------------------------------------------------------------------------ *)
 
+(*###TO Product_Type_lemmas.ML *)
 Goal "[|fst x = fst y; snd x = snd y|] ==> x = y";
 by (subgoal_tac "(fst x,snd x)=(fst y,snd y)" 1);
 by (rotate_tac ~1 1);
 by (asm_full_simp_tac(HOL_ss addsimps[surjective_pairing RS sym])1);
-by (asm_simp_tac (simpset_of Product_Type.thy) 1);
+by (asm_simp_tac (simpset_of (theory "Product_Type")) 1);
 qed "Sel_injective_cprod";
 
 Goalw [less_cprod_def] "(p::'a*'b) << p";